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Sample Variability. Consider the small population of integers {0, 2, 4, 6, 8} It is clear that the mean, μ = 4. Suppose we did not know the population mean and wanted to estimate it with a sample mean with sample size 2. (We will use sampling with replacement)

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sample variability
Sample Variability

Consider the small population of integers {0, 2, 4, 6, 8}It is clear that the mean, μ = 4. Suppose we did not know the population mean and wanted to estimate it with a sample mean with sample size 2. (We will use sampling with replacement)

We take one sample and get sample mean, ū1 = (0+2)/2 = 1 and take another sample and get a sample mean ū2 = (4+6)/2 = 5.

Why are these sample means different?

Are they good estimates of the true mean of the population?

What is the probability that we take a random sample and get a sample mean that would exactly equal the true mean of the population?

Section 7.1, Page 137

sampling distribution
Sampling Distribution

Each of these samples has a sample mean, ū. These sample means respectively are as follows:

P(ū = 1) = 2/25 = .08P(ū = 4) = 5/25 = .20

Section 7.1, Page 138

sampling distribution3
Sampling Distribution

Shape is normal

Mean of the sampling distribution = 4, the mean of the population

Section 7.1, Page 138

sampling distributions and central limit theorem
Sampling Distributions and Central Limit Theorem

Alternate notation:

Sample sizes ≥ 30 will assure

a normal distribution.

Section 7.2, Page 141

central limit theorem
Central Limit Theorem

Section 7.2, Page 144

central limit theorem6
Central Limit Theorem

Section 7.2, Page 145

calculating probabilities for the mean
Calculating Probabilities for the Mean

Kindergarten children have heights that are approximately normally distributed about a mean of 39 inches and a standard deviation of 2 inches. A random sample of 25 is taken. What is the probability that the sample mean is between 38.5 and 40 inches?

P(38.5 < sample mean <40) =

NORMDIST 1

LOWER BOUND = 38.5

UPPER BOUND = 40

MEAN =39

ANSWER: 0.8881

Section 7.3, Page 147

calculating middle 90
Calculating Middle 90%

Kindergarten children have heights that are approximately normally distributed about a mean of 39 inches and a standard deviation of 2 inches. A random sample of 25 is taken. Find the interval that includes the middle 90% of all sample means for the sample of kindergarteners.

Sampling Distribution

NORMDIST 2

AREA FROM LEFT = 0.05

MEAN = 39

ANSWER: 38.3421

NORMDIST 2

AREA FROM LEFT = .95

MEAN = 39

ANSWER: 39.6579

The interval (38.3 inches, 39.7 inches) contains the middle 90% of all sample means. If we choose a random sample, there is a 90% probability that it will be in the interval.

Section 7.3, Page 147

problems
Problems

Problems, Page 149

problems10
Problems

Problems, Page 150

problems11
Problems

Problems, Page 151

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